Another to throw onto the pile of people saying 'don't replace your camera just yet'. It sounds like there could be some experience issues that might be contributing to the issue more than the camera body, and you might be setting yourself up for some disappointment switching to a mid-range body and expensive lens only to discover the results aren't any better.
Shooting action involves special techniques, settings, and skill - but can be done with nearly any camera body...no need for mid-level cameras for that. The lens can play a part, depending on the light when you're shooting, and the camera body can sometimes help increase your odds or improve your controls...but they won't necessarily do any better than an entry level and can also do just as badly.
Post some sample shots, and include all the EXIF or camera settings for the shots...ISO, aperture, shutter speed, mode, focus area, focus mode, metering mode, metering area. Give some idea of the time of day, lighting, and a little of what you did when you shot (did you stand in one place and shoot in one direction, or move the camera along with the moving subject, did you half-press for focus and exposure lock first, then press to shoot, or just press the shutter all the way down, were you in a continuous burst mode or just pressing the shutter often, etc.
Though your camera isn't designed to be a pro-level sports camera, it can still capture action shots - with a lens sufficiently fast (bright) to deal with the lighting conditions, and a shutter speed of 1/500 or better, even a P&S camera can capture an animal in action with decent results. And what you want to achieve in an action shot can differ - do you want the subject isolated from the background and completely frozen, do you want some background motion blur to imply the movement of the subject, do you want everything crystal clear and in focus including the background, do you want the subject to have some blur at the feet or body to give the feeling of movement?
Sony DSLR-A580 / Sony 18-250mm / Minolta 50mm F1.7 / Sigma 30mm F1.4 / Tamron 10-24mm / Tamron 150-600mm / Tamron 90mm F2.8 macro / Minolta 300mm F4 APO
Sony A6000 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses